Shu-cheng S. Chen

Glenn Research Center, Cleveland, Ohio

A Guide to Axial-Flow Turbine Off-Design 

Computer Program AXOD2

NASA/TM—2014-218301

June 2014
	
	
	








	



	

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Shu-cheng S. Chen

Glenn Research Center, Cleveland, Ohio

A Guide to Axial-Flow Turbine Off-Design 

Computer Program AXOD2

NASA/TM—2014-218301

June 2014


	



	


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NASA/TM—2014-218301 1 

A Guide to Axial-Flow Turbine Off-Design 

Computer Program AXOD2 

Shu-cheng S. Chen

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135 

Summary 

A Users’ Guide for the axial flow turbine off-design computer program AXOD2 is composed. This 

Users’ Guide is supplementary to the original Users’ Manual of AXOD. Three notable contributions of 

AXOD2 to its predecessor AXOD, both in the context of the Guide or in the functionality of the code, are 

described and discussed in length. These are: 1) a rational representation of the mathematical principles 

applied, with concise descriptions of the formulas implemented in the actual coding. Their physical 

implications are addressed; 2) the creation and documentation of an ‘Addendum Listing’ of input 

namelist-parameters unique to AXOD2, that differ from or are in addition to the original input-namelists 

given in the Manual of AXOD. Their usages are discussed; and 3) the institution of proper stoppages of 

the code execution, encoding termination messaging and error messages of the execution to AXOD2. 

These measures are to safe-guard the integrity of the code execution, such that a failure mode encountered 

during a case-study would not plunge the code execution into indefinite loop, or cause a blow-out of the 

program execution. Details on these are discussed and illustrated in this paper. Moreover, this computer 

program has since been reconstructed substantially. Standard FORTRAN Langue was instituted, and the 

code was formatted in Double Precision (REAL*8). As the result, the code is now suited for use in a local 

Desktop Computer Environment, is perfectly portable to any Operating System, and can be executed by 

any FORTRAN compiler equivalent to a FORTRAN 90/95 compiler. AXOD2 will be available through 

NASA Glenn Research Center (GRC) Software Repository. 

1.0 Introduction 

AXOD is a preliminary axial-flow turbine off-design computer program, originally developed by the 

General Electric Company (Ref. 1) under contract for the NASA Lewis Research Center (now the NASA 

Glenn Research Center (GRC)). Since its delivery to GRC, a substantial amount of modifications, 

upgrades, and testing were performed on this code by researchers at the GRC. These efforts are reported 

in References 2, 3, and 4. A formal documentation of its past evolution, with extensive demonstration of 

its usages in aircraft turbine off-design performance analysis, was published in Reference 2. The code was 

named AXOD in that reference. A theoretical analysis and discussion on its mathematical foundation and 

formulations, with a systematic approach for the loss-model closure and the determination of the loss 

factors were described in Reference 3. Subsequently, a capability extension of AXOD to include the 

outlet guide vane (OGV) into its acceptable turbine configurations was reported in Reference 4. 

Evidently, these aforementioned works have generated a sufficient amount of interest in this computer 

program, that a formal update to the original Users’ Manual of AXOD composted in Reference 2 became 

preferable. The purpose of this work is to provide an updated guide to the description and the usage of the 

code. 

In this document, concise descriptions of the mathematics and the formulations of the code are 

presented. The bases of, and the underlying physics pertinent to, these formulas are discussed. An 

‘Addendum Listing’ of the input namelist parameters is provided. These input parameters are different 

from or are in addition to the original input-namelists given in the Manual of AXOD. The usages of these 

new parameters are addressed. A complete collection of the encoded program termination-messaging and 

error messages, that would or could be encountered in the execution of this code are displayed and 
NASA/TM—2014-218301 2 

discussed. Lastly, a sample of input file to this new code, and its outputs generated are enclosed and 

illustrated in the Appendices of this paper. 

Moreover, this computer program has since been reconstructed substantially. The use of standard 

Fortran langue was instituted, and the code was formatted in Double Precision (REAL*8). As the result, 

the code is now suited for use in a local Desktop Computer Environment, is perfectly portable to any 

Operating System, and can be executed by any Fortran Compiler equivalent to a Fortran-90 or Fortran-95 

compiler (Fortran-77 compiler was found problematic in handling the namelist-reads, thus was deemed 

unsuitable for this code). 

This new version of code is hereby renamed as AXOD2. 

2.0 Mathematics and Formulations 

The structure of code, the algorithm, the solution-seeking procedure, and the governing equations 

applied in AXOD2 have no difference to its predecessor AXOD. However, the bases of code 

construction, the principles of mathematics, the formulation of formulas and their underlying physical 

implications, have not been described and discussed in an orderly, rational manner in the past. In this 

section, the basics of AXOD2 code construction are described. The mathematical principles applied are 

discussed, and the formulas derived from these principles are listed in concise, detailed forms consistent 

with the forms encoded in AXOD2. Their physical implications are addressed. 

AXOD2 is constructed with the stator and the rotor as two separate entities. Stators are envisioned in 

the absolute frame-of-reference, while rotors are envisioned in the relative frame-of-reference. Between 

the stator and the rotor, an annulus region connects the two. Solutions are processed and obtained only at 

the starting and the ending stations of each component: the stator, the annulus, the rotor, and the annulus 

that follows. The information of the component losses needed to resolve the flow at a station is obtained 

via loss-modeling, as was described in References 2 and 3. The annulus regions, however, are treated as if 

there were no losses (isentropic) in the flow domain. The switching of the frame-of-references between 

the stator and the rotor is executed in the annulus region. The annulus region after the stator receives the 

flow conditions from the stator discharge in the absolute frame, and calculates the inflow conditions, in 

relative frame, for the rotor inlet. The annulus after the rotor receives discharge flow conditions in the 

relative frame from the rotor, and calculates the inflows needed, in absolute frame, for the following 

stator inlet. A schematic of the turbine stage is shown in Figure 1. 

Figure 1.—A schematic of one turbine stage and the numbering of stations 

0 1 1A 

2 2A 

Stator Rotor 
NASA/TM—2014-218301 3 

In Figure 1, station 0 denotes the stator inlet, station 1 denotes the stator discharge/annulus intake; 

station 1A denotes the annulus exit/rotor inlet, station 2 denotes the rotor discharge/annulus intake; and 

station 2A is the stage exit of the annulus. The radial flow domain is subdivided into several ‘sectors’, 

each with an equal radius-height. A maximum of 6 sectors is permissible. The number of sectors can be 

either even or odd; however an odd number of sectors are preferred and recommended. 

The flow of the turbine is circumferentially averaged over the ‘effective area’. At the stator inlet and 

the rotor inlet (stations 0 and 1A), this effective area is the blade-tip to blade-tip sector area, multiplied by 

the number of blade passages (i.e., the full area of the sector-annulus). At the stator discharge and the 

rotor discharge (stations 1 and 2), the effective area is the blade-tip to blade-tip area, subtracting the 

fraction contributed from the metal and viscous blockage of the blade, then multiplied by the total number 

of blade passages. The effective area of station 2A (the stage exit) is the full area of the annulus-exit. 

In the current structure of AXOD2, a dedicated subroutine is constructed for each of the four 

components. Again, these are: the stator, the annulus that follows, the rotor, and the annulus that follows 

the rotor. The turbine can be a multi-stage turbine, and with the work of Reference 4, the last turbine stage 

can be simply a half-stage with the rotor been “blanked-out”, transformed into an annulus-equivalent (this 

technique is described in length in Ref. 4). The acceptable turbine configuration of AXOD2 includes 

single- or multi-stage axial flow turbines, with an optional outlet guide vane (OGV) attached. 

The solution algorithm implemented in AXOD2 is a marching scheme. Computationally, flow 

solution is sought over the components from one discharge station to the next in sequence. In this process, 

all necessary inlet conditions of a component are always known from the calculation already been 

conducted at the discharge station of the preceding component (with the understanding that the flow 

conditions at the turbine inlet must be well-defined through input, to start this marching process.) Thus, 

the problem boils down to finding the solution at the current discharge station of interest. To obtain the 

flow solution at a station-of-discharge, typically a set of five first-principle (conservation) equations are 

solved for the set of five primary unknowns of the flow: the static pressure (P), the static temperature (T), 

and the three velocity components (Vx, Vu, Vr). The governing equations of the system of AXOD2 are 

presented and discussed herein. For clarity, these equations are cast in forms that are most consistent with 

the forms implemented in the coding of AXOD2. And as noted, the governing equations for the stator are 

cast in the absolute frame of reference; the equations for the rotor are cast in the relative frame of 

reference. 

2.1 Mass Conservation 

The equations for the mass flow rate at the discharge stations of a blade component are: 

	1 1 * Vx1  A1  1YCS 1  1 *Vm 1 cos 1 * A1 *  1YCS 1 (1S) 

2 2 * Wx 2 A2    1 YCR 2 2 *Wm 2 cos  2 * A2 * 1YCR 2 	        
 (1R) 

where, 	 denotes the mass flow rate through the discharge station; Vx and Wx denote, respectively, the 

absolute and the relative axial velocities of the flow at discharge; Vm and Wm denote, respectively, the 

absolute and the relative meridional flow velocities; A1 and A2 are the sector area of the annulus at the 

stator and at the rotor exit, and YCS and YCR are the area blockage loss factors of the stator and the rotor at 

discharge, respectively. The  and the
 denote the meridional flow angles (streamline slope angles) of 

the stator and the rotor, respectively. The definition of the meridional flow angle and its implication to 

AXOD2 are provided later in this section. 

The equation number with a ‘S’ denotes ‘stator’ in the absolute-frame; the equation number with a 

‘R’ denotes ‘rotor’ in the relative-frame. This notation is applied throughout this paper. Conservation 

equations applied to the annulus regions are similar to those for the stators and the rotors, except that all 
NASA/TM—2014-218301 4 

losses in an annulus region are zero. For example, at an annulus discharge, Equation (1S) or (1R) would 

apply but with the loss factor YC = 0. 

The continuity conditions, which are the actual enforcement of the conservation of mass, applied 

respectively to the stators and the rotors are: 

	      	  	

I I

r

I

I I I 0 w 1 1 * 0 1 (2S) 

	      	  	

I I

r A

I

A I I I 1 w 2 1 * 1 2 (2R) 

where, (wr1 – 1) denotes the fraction of coolant flow injected in between stations 0 and 1, with respect to 

the inlet total mass flow rate of the stator at station 0;(wr2 – 1) denotes the fraction of coolant flow injected 

in between stations 1A and 2, with respect to the inlet total mass flow rate of rotor at station 1A. The ‘I’ 

here is the sector-index. Notice that the continuity conditions over a blade component are enforced 

globally by summing up the sector mass flow rates. The total discharging flow rate is matched to the total 

incoming flow rate, including the coolant flow addition. 

Contrarily, the continuity conditions over the stations of the annulus region are: 

	1   wr 1A 1 *	1  	1A (3a) 

	2  (wr 2A 1) *	 2  	2A (3b) 

The flow rates are matched locally sector by sector and no summation is applied. The(wr 2A – 1) is the 

fraction of the coolant flow injected in the annulus region between stations 1 and 1A, with respect to the 

mass flow rate at the annulus intake (or equivalently, of the preceding stator discharge). The (wr 2A – 1) is 

the fraction of coolant flow injected in the annulus region between stations 2 and 2A, with respect to the 

flow rate at its annulus intake (or the preceding rotor discharge). 

In Equations (2S), (2R), (3a), and (3b), 	1 and 	2 were defined in Equations (1S) and (1R). Although 

not shown here, 	0, 	1A, and 	2A are defined the same way only without the blockage loss factor YC,

because these three stations are the stations of the blade-component-inlet (or equivalently, the stations of 

the annulus-discharge; see Fig. 1 for the illustration), and the flow is treated as frictionless there. 

2.2 Streamwise Momentum Conservation 

The streamwise momentum balance applied in AXOD2 is not a conservation equation, but rather it is 

merely a modeling equation for the loss of total pressure across a blade component. This streamwise 

momentum loss is casted into two explicit mechanisms, with two sets of loss coefficients YA and YB. 

The first loss mechanism is the stagnation region total pressure loss. The loss factors are expressed as: 

2

0

1

0

01

1

0

0

2

1 M

P

P

P

P YA t t

S 




 



  





































  















  







 (4S-1) 

2

1

1

1

1 2

1

1

1

2

1

A

A

t A

A

t A

R M

P

P

P

P YA 
 
 

 



  





























 





 
  







 





 
  







 (4R-1) 
NASA/TM—2014-218301 5 

Here, ‘01’ represents an interim state immediately after the stator inlet state 0 and ‘1A 2’ represents 

the interim state immediately after the rotor inlet state of 1A. The subscript (t) denotes the ‘total’ quantity. 

The superscript (') refers to the quantity in the relative frame. The  is the ‘specific heat ratio’, and the M

is the Mach number. These notations are applied consistently throughout this paper. 

Noted here, the precise formulas of the stagnation region total pressure loss implemented in AXOD2 

are actually in the forms of: 

 

















 






 



  





























 





  







 







!    







2

0

1

0

01

1

0

0

2

1 * 1 1 M

P

P

P

P YA t t

S opt

S (4S-2) 

 

















 


 
 

 



  





























 





 
  







 





 
 !    







2

1

1

1

1 2

1

1

1

2

1 * 1 1 A

A

t A

A

t A

R opt

R M

P

P

P

P YA (4R-2) 

The opt

S ! and the opt

R ! are the stagnation region total pressure recovery factors of the stator and the rotor, 

respectively. Since the flow Mach number operating in an aircraft turbine is mostly subsonic throughout 

the turbine flow path, the inlet stagnation pressure recovery factors would be unity (i.e., 1.0) in most of 

the turbine operations. An exception is when operating at the beyond-choke condition is required (e.g., 

when operated at the limit-loading condition). For that case, the stagnation pressure recovery factor at the 

inlet of an outlet guide vane (OGV) could be lower than one, but only slightly. For all studies conducted 

with AXOD2, the inlet stagnation pressure recovery factors, opt

S ! and opt

R ! , were set to unity. 

The second loss mechanism specified is the blade row kinetic energy loss. The loss coefficients are 

expressed as: 

















 









 





    id

t

id

t

t

t

S T

T T

T

T T YB

1

1 1

1

1 1 1 (5S) 























 









 











    id

t

id

t

t

t

R T

T T

T

T T YB

2

2 2

2

2 2 1 (5R) 

The superscript (id) refers to the ‘ideal’ quantities, and they are defined as: 















































































  

1

1

1

0

1

01

1

1 1 1 1 P

T

T T T P P P

t

t

t

id

t

id

t (6S) 





















































































 
  
  


1

2

1

1

2

1 2

1

2 2 2 2 P

T

T T T P P P

t A

t

t A id

t

id

t (6R) 
NASA/TM—2014-218301 6 

id

t T 1 is the ideal total temperature at the stator discharge, where the flow is assumed isentropically 

expanded in the stator from the interim state of ‘01’ to the state of 1; and id Tt 2


 is the ideal relative total 

temperature at the rotor discharge, where the flow is assumed isentropically expanded in the rotor from 

the interim state ‘1A 2’ to the discharge state of 2. Rigorous derivation for the two equations is omitted 

here, but notice that the discharge states of the actual quantities in these equations contain the total 

temperature ratios of 0

1

t

t

T

T and t A

t

T

T

1

2





 , respectively for the stator and for the rotor. This is the most general 

expression of the isentropic relation for the expansion process described here, and is the precise form 

applied in the coding of AXOD2. When no coolant flow is added, 0

1

t

t

T

T of the stator would be just unity 

(1.0); and if the sector meanline diameters at the rotor inlet and at the discharge were identical, the ratio 

t A

t

T

T

1

2





 of the rotor would also be one. However, when there is coolant flow addition, neither 0

1

t

t

T

T

 nor t A

t

T

T

1

2







should be unity, as they would be affected by the coolant temperature injected. 

In Equations (5S) and (5R), Tt1 and T1 are the actual absolute-total-temperature and the static 

temperature of the flow at the stator discharge; Tt2


 and T2 are the actual relative-total-temperature and the 

static temperature of the flow at the rotor discharge. These quantities are obtained by solving the system 

equations. 

Before moving on to the next conservation equation, an important point is made here. Although the 

streamwise momentum loss specified here consists of only two explicit mechanisms, however, solving the 

flow at a discharge station requires one to satisfy simultaneously the complete set of governing equations, 

which includes the mass conservation, the momentum conservations, and the energy conservation. Since 

the governing equations are inter-connected, therefore when counting the losses of streamwise total 

pressure (the streamwise momentum loss) across a blade-component from inlet to discharge, all three 

aforementioned loss factors: YA, YB, and YC (the area-blockage loss factor in the mass conservation) are 

contributing to the total loss. The effect of the blockage loss on momentum is implicit; it contributes the 

thermodynamic loss of the streamwise momentum through the changes in P (static pressure) and  

(density). This loss is in addition to the kinetic energy loss of YB. Another important point to be stated is 

that all three of the loss mechanisms are contributing visibly only to the streamwise momentum loss (the 

total pressure loss). The reason for this is that no other principal conservation equations described under 

the present context contains loss (viscous term) of the explicit nature. This statement will be clear as we 

move on to the discussion of the remaining principles of conservation. 

In application, the loss factors YA are correlated to the inflow incidence angles, the loss coefficients 

YB are correlated to a tangential blade loading factor, and the blockage loss factors YC are correlated to 

the discharge blade angles of the blade-component. These correlations are provided and discussed in 

detail in Reference 3. 

Equations (4S-1) and (4R-1), Equations (5S) and (5R), and Equations (6S) and (6R) are applicable to 

the annulus discharge but with the loss factors (YA, YB) set to zero, because the flow in the annulus 

regions are treated as isentropic. The actual states-of-equation of the Pt and the Tt applicable to an annulus 

discharge are derived directly from the equations of the absolute frame, i.e., Equations (4S-1), (5S), and 

(6S) with zero loss factors. Conversion of the quantities between the absolute-frame and the relativeframe, when needed, are done in a straight forward manner following their respective definitions. 

2.3 Angular Momentum Conservation 

The equations for the angular momentum conservation applied over the blade-components are: 

0  0 (7S) 
NASA/TM—2014-218301 7 

PW  " # U1A Wu 1A U1A U2 Wu 2 U2 GJ (7R) 

Where, ‘PW ’is the specific work-extract from the rotor (PW here is a positive quantity. The ‘-PW’ would 

be referred to as ‘the specific work-input’ or simply ‘the specific work’; the sign of PW in References 3 

and 4 was wrongfully presented). The U1A and U2 are the tangential rotor blade velocities, and Wu1A and 

Wu2 are the relative tangential flow velocities, at the rotor inlet (station 1A) and at the rotor discharge 

(station 2) respectively. The G and J are the unit conversion factors. 

Equation (7R) is commonly referred to as the Euler turbomachinery equation. It is the discrete form 

of the angular moment-of-momentum balance attributed to the change in angular momentum of the flow, 

from inlet to discharge over a blade-component. It is rigorously satisfied without the inclusion of the 

viscous term (loss), even though the flow across the blade-component is acknowledged to be viscous. 

Physically, for a turbine, it represents the net specific work extracted from the flow through the 

interaction with the blade component. This net work-output provides both the useful work (the shaft 

work) and the wasteful works (frictional losses), such as tip-clearance vortices formation, windage loss on 

the faces of the turbine disk, and bearing and seal loss, etc. Notice that Equation (7S) is simply an identity 

0=0. This is due to the fact that no work is produced over a stator, since it is stationary with no blade 

speed. 

The angular moment-of-momentum equation (Eq. (7R)) is a major governing equation for the design 

of turbomachinery, but ironically, for the off-design code such as AXOD2, this equation is in fact 

redundant. The reason for this is that the specific work-extract, PW, is a resultant to AXOD2, not a 

design-requirement, since AXOD2 is solving the direct problem, not the inverse problem as to a design 

code, in which ‘PW ’ would be a given quantity assigned from input. Although still connected implicitly 

to the system through the energy equation, Equation (7S) or (7R) on their own does not provide a 

meaningful relation in solving the unknowns (i.e., P, T, Vx, Vu, and Vr). For this system of governing 

equations to be well-posed (i.e., to be sufficient and unique in solving the unknowns), a set of additional, 

supplementary conditions must be provided at the stator and at the rotor discharge, respectively. In 

AXOD2, this additional condition is a specified (assigned) tangential flow angle at the blade-component 

discharge station. This treatment is described and discussed in length in the paper of Reference 4. The socalled ‘supplementary conditions of the velocity component equations’ are defined later in this section. 

In contrast, at the annulus discharge, the angular momentum conservation equations are well-defined 

and definitive. In the absence of the tangential blade force (no blades exist in the annulus region), they are 

simply given as: 

Vu 1 * D1 Vu 1A * D1A (7a) 

Vu 2 * D2 Vu 2A * D2A (7b) 

where, Vu is the absolute tangential-flow-velocity, and D is the sector meanline diameter. Subscript ‘1’ 

refers to station 1, and subscript ‘1A’ refers to station 1A. Subscript ‘2’ refers to station 2, and subscript 

‘2A’ refers to station 2A. 

Equations (7a) and (7b) replace Equations (7S) and (7R) of the blade-components, as the angular 

momentum conservation equations for the annulus discharge. Equations (7a) and (7b) are definitive and 

self-sufficient, no supplementary assignment is needed. When the expressions of the relative frame are 

needed, they are converted from these equations of the absolute frame in a straight forward manner. 

When the tangential flow angle is needed at the annulus discharge, it can be obtained analytically (as 

opposed to been assigned) by satisfying simultaneously the two equations: the mass conservation 

(Eqs. (3a) or (3b)), and the angular momentum conservation (Eqs. (7a) or (7b)). This derivation has some 

relevance to the work presented in Reference 4 and is given here: 

From Equations (3a) and (1S), one can write: 
NASA/TM—2014-218301 8 

wr1A*1 * A1 *Vx 1 *  1YCS 1  1A * A1A *Vx 1A (7c) 

And from Eq. (7a): 

Vu 1 * D1 Vu 1A * D1A

Dividing the two, we get: 

 




 







   A

A A

x r A S

u

x A

u A

D

D

A

A

V w YC

V

V

V

1

1

1

1

1

1

1 1 1

1

1

1 * * * 1

1 * 1 * (7d) 

Here, Vu is the absolute tangential-flow-velocity, Vx is the absolute axial-flow-velocity,  is density, A is 

the sector area of the annulus, and D is the sector meanline diameter. The wr1A is the ratio of the annulus 

discharge mass flow rate 	1A to the annulus intake mass flow rate 	1, seen in Eq. (3a). The YCS1 is the 

area blockage-loss factor at station 1 (in Eq. (1S)). The x

u

V

V above is the ‘tangent of the tangential flow 

angle’. 

The tangential flow angle at the discharge station 2A (i.e., at the stage exit), with respect to the 

annulus intake of station 2, is obtained the same way analogous to the derivation shown by Eqs. (7a), 

(7c), and (7d). For this, one obtains, 

 




 







   A

A A

x r A R

u

x A

u A

D

D

A

A

V w YC

V

V

V

2

2

2

2

2

2

2 2 2

2

2

2 * * * 1

1 * 1 * (7e) 

When the relative tangential flow angle is needed, it can be deduced by simply converting from the 

absolute tangential flow angle expressed here. 

2.4 Radial Momentum Conservation 

The equations of radial momentum conservation (the radial equilibrium conditions) applied in 

AXOD2 are: 

r

V

dr

G dP u 2

1 1

1

 

 (8S) 



r

W U

dr

G dP u 2

2 2 2

2

  

 (8R) 

The expression is commonly referred to as the ‘simple radial equilibrium condition’. An elaborated 

derivation of this equation can be found in Reference 5. P1 and P2 are the static pressures, 1 and 1 are 

densities, at stator discharge and at rotor discharge, respectively. Vu1 is the absolute tangential flow 

velocity at a stator discharge, Wu2 is the relative tangential flow velocity at the rotor discharge. The U2 is 

the blade velocity at the rotor discharge. The r is the radial coordinate. 

The radial equilibrium condition is a differential equation; it merely provides the distribution relation 

of P along the radial direction at a discharge station. Although extremely important for the flow solution 

to be physical, but no absolute magnitude of P can be obtained from this equation alone. It has to be 

coupled with the equation of mass conservation and solved together, until the continuity condition (Eq. 

(2S) or (2R)) is satisfied over the blade-component. Thus the coupling of Equations (8S) and (1S), or 

Equations (8R) and (1R), together constitutes only one well-posed relation for the system at the stator 
NASA/TM—2014-218301 9 

discharge, or at the rotor discharge. Therefore, once again, we are facing with the problem of an ill-posed 

system; the flow solution is not obtainable unless a set of supplementary conditions are given at the stator 

exit and at the rotor exit, respectively. In AXOD2, this set of supplementary conditions is the 

specifications (assignments) of the meridional flow angles (streamline slope angles) at the stator and at 

the rotor discharge, respectively. These assignments supplement the deficiency of the need to couple the 

two principal equations: Equations (8S) and (1S) at the stator discharge and Equations (8R) and (1R) at 

the rotor discharge. This argument is also discussed in Reference 4. 

In contrast, at the annulus discharge the radial equilibrium condition is not applied at all. This is 

because the continuity condition (the conservation of mass) at the annulus discharge is enforced locally 

sector-by-sector as was described by Eqs. (3a) and (3b). Thus it is self-sufficient at the annulus discharge. 

The coupling of mass and the radial pressure distribution is unnecessary. In AXOD2, the well-posedness 

of the system at the annulus discharge is ensured simply through the adaptation of the meridional flow 

angles (streamline slope angles) being given there (assigned; equal to the streamline slope angles at their 

respective annulus intakes.) The radial distribution of pressure is in fact ignored at the annulus discharge. 

2.5 The Energy Conservation 

The last principle of conservation is the conservation of energy. The equations implemented in 

AXOD2 are the followings: 

 1 1 1

cool

01

cool 0 0 1 01

* 1 * * * *

t r t r t cp T  w  cp T  w cp T (9S) 

" #    2 2 2

cool

1 2

cool 1 2 1 2

2

1

2 1 1 2

* 2 * 1 * *

A r * *

t A A A r A t cp A Tt


  U U GJ cp  w  cp T
  w cp T
 (9R-1) 

Whereas in Eq. (9R-1), 

" #  cool

1 2

2

2

cool

1 2

cool

1 2 2 *

t A t A A T
  T U GJ cp (9R-2) 

In Equation (9S), Tt0 andTt1 are the absolute total temperatures of the flow at the stator inlet and at its 

discharge, respectively. Them cool

t 01 T is the absolute total temperature of the coolant injected into the 

stator passage, between station 0 and station 1. The (wr1 – 1) denotes the mass fraction of the coolant flow 

addition, with respect to the mainstream mass flow rate at the stator inlet of station 0. The cp’s are the 

specific heats of the media at constant pressure. In Equation (9R-1), Tt1A
 and Tt2


 are the relative total 

temperatures of the flow at the rotor inlet and at the rotor exit, respectively. The cool

t 1A2 T
 is the relative total 

temperature of the coolant injected into the rotor passage between station 1A and station 2. The (wr2 – 1) 

is the fraction of the coolant mass flow injected, with respect to the inlet mass flow rate of the rotor at 

station 1A. And cp’s are the specific heats of the media at constant pressure, at various locations. The U1A

and U2 are the blade speeds (tangential blade velocities) at the rotor inlet and at the rotor discharge, 

respectively. 

In Equation (9R-2), cool

t 1A2 T
 is the relative-total-temperature and cool

t 1A2 T is the absolute-totaltemperature of the coolant flow injected into the rotor passage. This equation is formulated with the 

argument of compatibility to the right-hand-term in Eq. (9R-1). 

The energy conservation equations for the blade-components expressed here can be derived 

rigorously. But the processes, in particular those for the energy equation of the rotor, are quite involved. 

This detailed derivation is omitted here. 

The energy equations for the annulus regions, including the coolant flow additions, are implemented 

in the manner analogous to Eq. (9S), i.e., the equation of the absolute frame. The relative total 
NASA/TM—2014-218301 10 

temperature of the flow at the annulus discharge, when needed, is simply deduced out from the absolute 

total temperature of the flow at discharge, obtained in analogy with Eq. (9S). 

Note that, the formulas given above are cast in forms consistent with the forms implemented in the 

coding of AXOD2, with a minor difference that the coefficients of specific heat (the cp’s) in AXOD2 are 

mass-averaged between the two corresponding states. This averaging is somewhat difficult to be included 

here without cluttering the expressions of the energy equations. Thus, the present forms of expression are 

adopted. 

The derivation of the equations for the energy conservation presented here involves the assumptions 

of: 1) frictionless flow at the end-stations (i.e., the inlet and the discharge stations) of a component; 2) 

insulated (adiabatic) walls; and 3) the effect of the radial flow mixing in the component passage is 

ignored (dismissed). 

2.6 Supplementary Conditions of the Velocity Component Equations 

In addition to the five principal equations, a set of supplementary velocity component equations are 

needed. 

For the stators: 

2 2 V  Vm Vu (10a) 

2 2 Vm  Vx Vr (10b) 

 Vu Vx 1 tan $  (10c) 

tan , 1 Vr Vx    (10d) 

and thus, 

Vx Vm * cos   (10e) 

Vr Vm * cos    * tan  (10f) 

And for the rotors: 

2 2 W  Wm Wu (11a) 

2 2 Wm  Wx Wr (11b) 

 Wu Wx 1 tan $
  (11c) 

tan , 1 Wr Wx  
  (11d) 

and thus, 

 m * cos (
) Wx W (11e) 

 m * cos (
) * tan (
) Wr W (11f) 

In here, V and W are the absolute and relative total velocities of the flow, Vm and Wm are the absolute 

and relative meridional velocity components, Vx and Wx are the absolute and relative axial velocity 

components, Vu and Wu are the absolute and relative tangential velocity components, and Vr and Wr are the 

absolute and relative radial velocity components. The $ and $
 are the tangential flow angles at stator and 
NASA/TM—2014-218301 11 

at rotor discharge, respectively. The  and 
 are the meridional streamline slope angles of the stator and 

the rotor, respectively. 

As noted previously, the $ and $
 and the  and 
 are the supplementary conditions necessary for the 

well-posedness of the system equations. They are always assigned at the discharge stations of the stator 

and the rotor. The manner in which they are assigned is described in Reference 4. 

In AXOD2, the flow direction is always placed onto the meridional plane, instead of the axial plane. 

The major velocity components used in AXOD2 are the Vm and Wm and the Vu and Wu. When Vx and Wx

or Vr and Wr are needed, they are expressed in form of Equations (10e) and (11e) or (10f) and (11f), 

respectively. 

This concludes the section of ‘Mathematics and Formulations’. 

Before moving on to the next subject, it is noted here that the solution processing algorithm 

implemented in AXOD2 is well developed but rather complex. The step-by-step solution-seeking 

procedure can be tracked more easily through the documentation and the code-descriptions provided in 

Reference 1. However, many of the detailed treatment that currently implemented in code AXOD2 are 

substantially different from the coding described in Reference 1. The step-by-step processing of the 

algorithm in solving the system equations is not included in the scope of this paper, and no attempt shall 

be made to describe it here. For the interested reader, to be proficient with the codes encoded in AXOD2, 

one has to get into it and read/check the statements line-by-line, with Reference 1 serves as a guide or 

overview for the understanding of the flow chart and logic of this code. 

In regard to the practical application of this code, there are many validation cases performed and the 

results were reported comprehensively in Reference 2, 3, and 4. The procedure undertaken in the usage of 

the code for problem solving has also been documented, illustrated, and reported in these references. This 

subject will not be repeated here, interested reader is referred to these papers cited. 

3.0 Addendum Listing of the Input Namelist Parameters 

Through modifications and upgrades described in References 3 and 4, some of the original input 

parameters in code AXOD were redefined, and some new ones were added. However, care was duly 

taken in AXOD2 to ensure these modifications and upgrades do not interfere with or altering the original 

functionality of the code. A list of input parameters that differ from or are in addition to the original 

namelist input described in the Users’ Manual of AXOD (Ref. 2) is provided herein. 

1. ‘KOGV’ (scalar; = 0 or 1); a new parameter: 

When KOGV=1, the Stator of the "KSTG" stage (the last stage) is the outlet guide vane (OGV); 

the Rotor of the "KSTG" stage is blanked-out. 

When KOGV=0, all stages are in the full-stage configuration. 

Default is "KOGV=0". 

2. Add ‘cxs, cxSR, cxr, cxRS’ (scalars; inches) to the input namelists: 

‘cxs’ is the axial length of the stator vane, 

‘cxSR’ is the axial length of the annulus of stator-rotor, 

‘cxr’ is the axial length of the rotor blade, 

‘cxRS’ is the axial length of the annulus of rotor-stator. 

No default is assigned to ‘cxs, cxSR, cxr, cxRS’. 

With these user-input lengths, one can now accurately specify the axial dimensions of the turbine 

stages, which are used to calculate the sector slopes for the meridional flow angles (Equations 10(d) and 

11(d) cited in this paper). 
NASA/TM—2014-218301 12 

To engage this option of user-specified axial lengths, one needs to set "iar=0" in the namelist-input. 

And needs to provide the data for ‘cxs, cxSR,cxr,cxRS’ stage-by-stage in the input namelists. 

When "iar" is not zero, the vane/blade/annulus axial lengths are determined via empirical sizing 

correlations cited by Glassman (Ref. 6), which are built-in to the code. These options are unchanged and 

are still active as they were. The original "iar=0" option (pure axial turbine), however, has been replaced. 

 Default for "iar" is 2 (medium aspect ratio blades). 

3. Add ‘ANGminS(6)’ (degrees) to the input namelists; 

 Add ‘ANGminR(6)’ (degrees) to the input namelists. 

 ‘ANGminS’ is a 1-D array with the dimension of sectors employed. 

Default is 0.0. 

It specifies the inlet stagnation region streamline deflection angle (due to flow circulation; in degrees) 

at each sector of the Stator. 

 ‘ANGminR’ is a 1-D array with the dimension of the sectors employed. 

Default is 0.0. 

It specifies the inlet stagnation region streamline deflection angle (due to flow circulation; in degrees) 

at each sector of the Rotor. 

If assigned, these parameters are given/read stage-by-stage. But with certain "SLI" options, automatic 

matching is made, so that only some of them are the required input. Auto-matching is conducted on these 

parameters according to the similarity relation cited in Reference 3. 

4. Reformulated ‘SETA(6)’ (dimensionless) in the namelist-input; 

Reformulated ‘RETA(6)’ (dimensionless) in the namelist-input. 

 ‘SETA’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 

It specifies the value of the blade-row efficiency of the Stator (the (1-YBS) cited in Equation (5S) of 

this paper). 

 ‘RETA’ is a 1-D array with the dimension of the sectors employed. 

Default is 1.0. 

It specifies the value of the blade-row efficiency of the Rotor (the (1-YBR) cited in Equation (5R) of 

this paper). 

If assigned, these parameters are given/read stage-by-stage. But with certain "SLI" options, automatic 

matching is made, so that only some of them are the required input. Auto-matching is conducted on these 

parameters according to the similarity relation cited in Reference 3. 

5. Reformulated ‘SCF(6)’ (dimensionless) in the namelist-input; 

 Reformulated ‘RCF(6)’ (dimensionless) in the namelist-input. 

 ‘SCF’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 
NASA/TM—2014-218301 13 

It specifies the value of the blade-row area blockage factor of the Stator (the (1-YCS) cited in Equation 

(1S)). 

 ‘RCF’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 

It specifies the value of the blade-row area blockage factor of the Rotor (the (1-YCR) cited in Equation 

(1R)). 

If assigned, these parameters are given/read stage-by-stage. But with certain "SLI" options, automatic 

matching is made, so that only some of them are the required input. Auto-matching is conducted on these 

parameters according to the similarity relation cited in Reference 3. 

6. Reformulated ‘SREC(6)’ (dimensionless) in the namelist-input; 

 Reformulated ‘RREC(6)’ (dimensionless) in the namelist-input. 

 ‘SREC’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 

It specifies the baseline value of the stagnation region total pressure recovery factor of the Stator, 

cited in Equation (4S-2). 

 ‘RREC’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 

It specifies the baseline value of the stagnation region total pressure recovery factor of the Rotor, 

cited in Equation (4R-2). 

‘SREC’ and ‘RREC’ are the reserved parameters for the stagnation region total pressure loss, in 

addition to the loss contributed from the flow incidence effect. As noted in Section 2.0 of this paper, for 

all cases studied, the value of the stagnation region total pressure recovery factors were set at 1.0 (the 

default; no baseline loss of stagnation pressure at the stator or at the rotor inlet). 

The usage of this option has not been explored, and can be reformulated in the code to account for the 

specific effect on losses, such as the shock loss or the unsteadiness loss, when necessary. 

7. Add ‘STF(6)’ (dimensionless) to the input namelists; 

Add ‘RTF(6)’ (dimensionless) to the input namelists (actually ‘RTF’ already exists as a namelistinput in the original code, but is redefined here). 

 ‘STF’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 

It is an “extra loss factor” multiplied to the blade-row loss coefficient (the YBS) of the Stator. 

 ‘RTF’ is a 1-D array with the dimension of the sectors employed. 

 Default is 1.0. 

It is an “extra loss factor” multiplied to the blade-row loss coefficient (the YBR) of the Rotor. 

‘STF’ and ‘RTF’ are strictly user-specified parameters, no auto-matching are made on these. When 

assigned, these two parameters are to be given/read sector-by-sector, and stage-by-stage. If a particular 
NASA/TM—2014-218301 14 

blade row is not being given, that would mean the STF (/RTF) is 1.0 (the default) for that blade row. 

These factors need not be given again at the next speed-line; the same value will be automatically applied. 

The similarity laws cited in Reference 3 implicitly assume the proportionality is with a constant 

coefficient. These “extra loss factors” are applied when the losses in a rotor or a stator are known 

different from the similarity relation of constant coefficient. 

8. The ‘SLI’ options: 

The option of ‘SLI’ determines how the following parameters are to be specified or be interpreted: 

a. Stagnation region streamline deflection angles: ANGminS, ANGminR; 

b. Blade-row efficiencies: SETA, RETA; 

c. Area blockage factors: SCF, RCF; 

d. Stagnation region total pressure recovery factors: SREC, RREC. 

 SLI = -1, 1 are the two original options in the code. 

 SLI = 0, 2, 3, 4 are the newly added auto-matching. 

Default is "SLI=1" (to preserve the execution/result of the original cases). 

The most practical usage is "SLI=3". 

SLI = -1 : All sectors and all blade-rows need to be given/assigned. 

SLI = 0 : Need input for each sector of both the first stator and the first rotor. 

The remaining stators of the ensuing stages will be determined automatically based 

on the similarity relations with respect to the first stator; 

The remaining rotors of the ensuing stages will be determined automatically based 

on the similarity relations with respect to the first rotor. 

SLI = 1 : Need input for each sector of both the first stator and the first rotor. 

The remaining stators of the ensuing stages will be assigned the value exactly 

identical to the value of the first stator. No similarity law is applied. 

The remaining rotors of the ensuing stages will be assigned the value exactly 

identical to the value of the first rotor. No similarity law is applied. 

SLI = 2 : Need input on the meanline sector of the stator-only for each stage. 

Within each stage, the sectors will be auto-matched to the value assigned to the 

meanline sector, according to the similarity law; the rotor will be auto-matched to 

the value of the stator, again, utilizing the similarity law. 

This process is conducted stage-by-stage, thus this input (on the meanline sector of 

the stator) needs to be provided at each stage. 

SLI = 3 : Need input on the meanline sector of the first stator only. 

All sectors and all blade-rows are auto-matched to the value assigned to the 

meanline sector of the first stator, utilizing the similarity law. 
NASA/TM—2014-218301 15 

This option is the most robust option available. And it was applied to all the newly 

studied cases conducted with AXOD2. 

SLI = 4 : Same as SLI=3, only the radial sectors of the first stator allow/require separate 

user input, instead of applying the similarity relation onto the sectors. 

9. Modified ‘PAF’ (dimensionless) in the namelist-input: 

When “PAF=0.”: uniform, mass-averaged radial profile of the total temperature (Tt), and an 

entropy-averaged radial profile of the total pressure (Pt) are seen at the inlet of 

the ensuing stage (a complete mixing is assumed to occur at the stage exit). 

When “PAF=1.”: the radial profiles of the Tt and the Pt seen at the inlet of the ensuing stage 

remain the same as those obtained at the annulus-exit of the current stage (the 

stage exit-flow is completely un-mixed). 

When “0 < PAF < 1”: the radial profiles of both the Tt and the Pt seen at the inlet of the ensuing 

stage are the weighted-average between the profile of “PAF=0” and the 

profile of “PAF=1” (partial mixing at the stage exit is assumed). 

 Default is “PAF=0.” 

10. Add ‘WAIRcool’ (a scalar; dimensionless) as an input parameter. 

‘WAIRcool’ is the water-to-air mass ratio specific for the coolant mixture. 

Coolant was specific to just air, it can now be the mixture of water-vapor and air. 

 Default is 0. 

11. The options for the plot-file: iplot(3) 

‘iplot’ is an array of dimension 3. 

 iplot(1) = 1: Equivalent mass flow rate (lbm/s) at the turbine inlet. 

 = 2: Corrected mass flow rate (lbm/s) at the turbine inlet. 

 = 3: The plain mass flow rate (lbm/s) at the turbine inlet. 

 = 4: Equivalent mass flow rate (lbm/s) at the first rotor (rotor 1) inlet. 

 = 5: Corrected mass flow rate (lbm/s) at the first rotor (rotor 1) inlet. 

 = 6: The plain mass flow rate (lbm/s) at the first rotor (rotor 1) inlet. 

 = 7: The overall turbine power produced (horse-power). 

 = 9: The overall torque (ft-lb). 

 iplot(2) = 1: The total efficiency (overall). 

 = 2: The static efficiency (overall). 

 = 3: A rating efficiency (overall; an efficiency which considered the kinetic energy of the 

exit swirling to be a loss). 

 iplot(3) = 1: The overall total-to-total pressure ratio. 

 = 2: The overall total-to-static pressure ratio. 

 = 3: The overall blade-jet speed ratio. 

 Default is "iplot = 1,1,1," 

This concludes the supplementary listing of the input namelist-parameters unique to AXOD2. 
NASA/TM—2014-218301 16 

Some remarks in regarding the output from AXOD2: 

A considerable amount of effort was afforded to the printouts of AXOD2. The variables deduced in 

the output routines (both the stage-by-stage, and the inter-stage sector-by-sector printouts) were carefully 

examined for their adequacy and correctness. As the result, some clarifications were made on the stationindex where the variable was obtained, and some modifications were made concerning ambiguity of the 

definition or the deduction of a quantity on a few of the printed output. No serious error was found. A few 

new quantities were added to enhance the completeness of the output, where information was available 

but was not printed previously. These new additions however are quite apparent, and warrant no 

particular introduction. The original output listing provided in the Manual of AXOD (Ref. 2) remains 

sufficient. No update on the printout-listing is offered nor needed. 

4.0 Termination Messaging and the Error Messages 

The institution of proper stoppages and the messaging for the termination of code execution is also 

considered to be a notable contribution of AXOD2 to its predecessor. A substantial amount of effort was 

spent to ensure the integrity of the code execution is in a sound state. Potential failure modes are 

identified, and proper messages are provided before the stoppage of the code execution, which otherwise 

would or could plunge the code into infinite loop. A summary of these incidences and the messaging are 

described herein. These messages are printed in the Standard Output (unit 6) of AXOD2. 

AXOD2 code-execution would terminate on two circumstances: natural termination (with full output 

given) and abrupt termination (with no output given). They are described herein: 

4.1 Natural Termination 

1. In the sweeping mode, the code terminates at the condition of last-rotor-choked, when “DELA=0” is 

specified in the input-namelist. The message would look like: 

 CASE 1 POINT 66 

 BLADE ROW 4 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.94570 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.66500 

 CASE 1 POINT 66 

 BLADE ROW 4 CHOKED. (CHOKE-ITERATION COMPLETE) 

 PT/PS of the starting blade-row is now 1.66322 

 PT/PS of this blade-row is now 1.94170 

 *** CASE 1 IS DONE *** 

2. In the sweeping mode, the code terminates at the limit-loading condition, when “DELA>0” is 

specified in the input-namelist. The message would look like: 

 CASE 2 POINT 53 

 BLADE ROW 8 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.97108 

 All Re-Iterations Begin from BLADE ROW 1 
NASA/TM—2014-218301 17 

 PT/PS at the Starting Blade-Row is 1.36000 

 CASE 2 POINT 53 

 BLADE ROW 8 CHOKED. (CHOKE-ITERATION COMPLETE) 

 PT/PS of the starting blade-row is now 1.35917 

 PT/PS of this blade-row is now 2.08400 

 CASE 2 POINT 53 

 BLADE ROW 9 CHOKING-DETECTED. 

 PT/PS of this blade-row is 2.21485 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.35917 

 BLADE ROWS 8 AND 9 CHOKE-DETECTED 

 Re-Start this POINT from BLADE ROW 1 

 DELPR at that Blade Row Was 0.50000E-02 

 DELPR Will Now Be 0.10423E-02 

 PT/PS of the starting value is 1.35604 

 CASE 2 POINT 55 

 BLADE ROW 10 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.94428 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.35813 

 CASE 2 POINT 55 

 BLADE ROW 10 CHOKED. (CHOKE-ITERATION COMPLETE) 

 PT/PS of the starting blade-row is now 1.35760 

 PT/PS of this blade-row is now 2.01263 

 POINT 69 

 Limit Loading is Reached. AACS = 1.0000 

 *** CASE 2 IS DONE *** 

3. Also in the sweeping mode, the code terminates at the outlet-guide-vane-choke, when both 

“DELA>0” and “KOGV=1” are specified in the input-namelist, and the outlet guide vane (OGV) 

happens to be found choked. The message would look like: 

 CASE 3 POINT 45 

 BLADE ROW 8 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.81833 
NASA/TM—2014-218301 18 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.38800 

 CASE 3 POINT 45 

 BLADE ROW 8 CHOKED. (CHOKE-ITERATION COMPLETE) 

 PT/PS of the starting blade-row is now 1.38790 

 PT/PS of this blade-row is now 1.91344 

 POINT 81 

 The OGV is Found Partially Choked. 

 Upstream Flow Rate Can Not Go Through. 

 The Last Data Point is Contaminated. 

 *** CASE 3 IS DONE *** 

4. In the single-point mode, when “WG” (mass flow rate at the turbine inlet) is assigned through the 

namelist-input, the code terminates when the solution is obtained and the output is given. The 

message is simply: 

 *** CASE 1 IS DONE *** 

Definition of the limit-loading condition and the circumstance of the OGV found-choked, as indicated 

above, are described and discussed in length in Reference 4. Reader is referred to that paper for detail. 

4.2 Abrupt Termination 

The abrupt termination occurs when a failure mode is encountered during the code execution. 

Execution is simply stopped on the spot, error message is given, and no output is available at this 

incidence. 

1. “WG” specified is too large for the single-point mode of execution. The message is: 

 CASE 1 POINT 1 

 BLADE ROW 4 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.87398 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.33839 

 SINGLE POINT MODE. WG = 65.0000 LBM/S 

 EXECUTION TERMINATED. NO OUTPUT IS GIVEN. 

 WG is out-of-range. Try A Smaller WG. 

2. “WG” specified is too small for the single-point mode of execution. The message would be: 

UNDER-FLOW in SUBROUTINE STA1. 

OUTPUTS ARE MEANINGLESS at THIS CONDITION. 
NASA/TM—2014-218301 19 

PROGRAM IS TERMINATED. TRY A HIGHER PTPS. 

Although the instruction here is “TRY A HIGHER PTPS”, but in this incidence, the root cause is 

actually “WG” specified being too small, which caused the flow at station 1 to be under-developed. 

One should try using a larger “WG”. 

3. Under-flow/reverse-flow detected. This occurs when executing in the sweeping mode and the “PTPS” 

value specified in the input-namelist is too small (too close to 1.) for the flow to develop in the 

turbine flow path. This message may look like: 

 CASE 1 POINT 1 

 BLADE ROW 7 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.00936 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.03800 

 CASE 1 POINT 1 

 BLADE ROW 7 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.00890 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.03600 

 CASE 1 POINT 1 

 BLADE ROW 7 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.00793 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.03400 

 CASE 1 POINT 1 

 BLADE ROW 7 CHOKING-DETECTED. 

 PT/PS of this blade-row is 1.00751 

 All Re-Iterations Begin from BLADE ROW 1 

 PT/PS at the Starting Blade-Row is 1.03200 

 UNDER-FLOW in SUBROUTINE STA1. 

 OUTPUTS ARE MEANINGLESS at THIS CONDITION. 

 PROGRAM IS TERMINATED. TRY A HIGHER PTPS. 

At this incidence, the cause is indeed “PTPS” specified being too small, one should try using a larger 

“PTPS” to start the sweeping mode execution. 

There are two other possible scenarios where an abrupt termination would occur, however neither of the 

two incidences has been experienced so far. Nevertheless, they are prepared in the code. These are: 

4. An intermediate blade row is found alone-choked in the turbine flow path. This message would be 

given in MAIN of the program. In its exact Fortran statements, it writes: 

 write(6,2015) JBRC 
NASA/TM—2014-218301 20 

 write(9,2015) JBRC 

 2015 format(17x,'Blade Row ',I2,' is Found Alone-Choked.'/, 

 1 17x,'Last-Blade-Row-Choke is Not Reachable.'/, 

 1 17x,'A Design Flaw Exists in The Turbine Flow Path.'/, 

 1 17x,'PROGRAM IS TERMINATED.',//) 

5. A two-blade-row-found-choked scenario has occurred in the turbine flow path, and “DELPR” is 

below the mark of 5.E-8 (near machine zero; the machine accuracy of the desktop computer used for 

the code development was determined to be at 1.E-8). “DELPR” is the pressure ratio incremental 

change (increase) in a single iterative step. This message would occur in SUBROUTINE LOOP, and 

in the exact Fortran statements it writes: 

 write(6,2010)IPC2,IBRC 

 write(9,2010)IPC2,IBRC 

 2010 format(17X,'Blade Rows ',I2,' and ',I2, 

 1 ' Are Choked Simultaneously.'/, 

 1 17x,'(DELPR < 5.E-8). A Design Flaw Exists in The Flow Path.'/, 

 1 17x,'PROGRAM IS TERMINATED.',//) 

This completes the section of ‘Termination Messaging and The Error Messages’. And as stated in the 

beginning of this section, it is believed that all possible scenarios for a failure mode to occur in the code 

execution have been accounted for. This ensures the integrity of the code execution, that a blow-out or 

plunging into indefinite loop will not happen in a case study. 

5.0 Concluding Remark 

A Users’ Guide for the computer program AXOD2 is provided. This Users’ Guide is supplementary 

to the original Users’ Manual of AXOD. Three notable contributions of AXOD2 to its predecessor 

AXOD were described and discussed in this paper. Firstly, the governing equations of the system were 

listed in a concise manner, that are consistent with the forms actually encoded in AXOD2. Rational 

representation and the implication of the mathematical principles applied were discussed. Secondly, a 

documentation of the ‘Addendum Listing’ of input namelist-parameters unique to AXOD2, that are 

different from (redefined) or are in addition to the original input-namelists given in the Manual of AXOD 

was composed. The functionality and the usages of these new input parameters were addressed. And 

thirdly, the institution of proper stoppages of the code execution in AXOD2, with their corresponding 

termination messaging and/or error messages were described and displayed. Moreover, this computer 

program has been reconstructed substantially from its original form of AXOD. Standard Fortran Langue 

was instituted, and the code is formatted in Double Precision (REAL*8). As the result, the code is now 

suited for use in a local Desktop Computer Environment, is perfectly portable to any Operating System, 

and can be executed by any Fortran Compiler equivalent to a Fortran-90/95 compiler. AXOD2 will be 

available through NASA Glenn Research Center (GRC) Software Repository. 
NASA/TM—2014-218301 21 

Appendix A.—Sample Input 

The sample input-file provided here is a 4 1⁄2-stage turbine with an average stage loading factor of 

4.66, designed by Pratt & Whitney Aircraft (Ref. 7) and experimentally tested in the turbine rig at NASA 

Glenn Research Center (Ref. 8). This case was studied with AXOD2 and reported in Reference 4. It is 

utilized here as a sample case. The input file reads: 

 NASA/P&W (Four and 1/2)-Stage Fan-Drive Turbine (ref. TMX-3498) 

 100 PERCENT SPEED (MUST GO FIRST) 

 &DATAIN 

 TTIN=518.7,PTIN=14.7, 

 PTPS=1.35,DELC=.002,DELL=.01,DELA=0.05, 

 STG=5.,SECT=5.,PCNH=5*0.2,AACS=1.0,VCTD=1., 

 RG=53.35,RPM=2980., 

 GAMG=5*1.4,RWG=5*1.0, 

 iplot=3,1,1, 

 SLI=3.0,IAR=0,ICYL=1,icf=1,EPR=1.0, 

 EXPN=4.,EXPP=3., 

 STAGE=1.,ENDJOB=0.,endstg=0.,endplt=1., 

 DR=15.956,15.700,15.538,15.538,15.458, 

 DT=19.000,19.280,19.596,20.154,20.492, 

 cxs =0.6, 

 cxSR=0.2, 

 cxr =0.857, 

 cxRS=0.286, 

 SDIA=5*0., 

 SDEA=59.95,62.43,64.27,65.47,66.03, 

 RDIA=50.83,53.53,55.20,55.85,55.47, 

 RDEA=64.46,66.22,67.34,67.82,67.66, 

 SETA=5*.956, 

 scf=5*0.984, 

 ANGminS=5*3.0, 

 stf=5*1.0,RTF=5*1.0, 

 SREC=5*1.0,RREC=5*1.0, 

 &END 

 &DATAIN STAGE=2., 

 RWG=5*1.0,GAMG=5*1.4, 

 DR=15.458,15.318,15.180,15.180,15.120, 

 DT=20.492,21.030,21.328,21.964,22.282, 

 cxs =0.992, 

 cxSR=0.331, 

 cxr =1.058, 

 cxRS=0.353, 

 SDIA=57.12,58.67,59.06,58.29,56.35, 

 SDEA=64.68,66.85,68.37,69.25,69.48, 

 RDIA=57.45,59.20,59.71,58.96,56.97, 

 RDEA=65.75,67.51,68.63,69.11,68.95, 

 endstg=0., 

 &END 

 &DATAIN STAGE=3., 

 RWG=5*1.0,GAMG=5*1.4, 

 DR=15.120,14.842,14.662,14.662,14.582, 

 DT=22.282,22.920,23.218,23.874,24.172, 

 cxs =1.134, 

 cxSR=0.378, 

 cxr =1.295, 

 cxRS=0.432, 

 SDIA=58.58,59.76,59.70,58.39,55.84, 
NASA/TM—2014-218301 22 

 SDEA=64.51,66.69,68.22,69.09,69.31, 

 RDIA=55.6,56.3,55.9,54.3,51.0, 

 RDEA=64.0,65.6,67.2,67.2,67.2, 

 endstg=0., 

 &END 

 &DATAIN STAGE=4., 

 RWG=5*1.0,GAMG=5*1.4, 

 DR=14.582,14.224,14.006,14.006,14.006, 

 DT=24.172,24.770,25.068,25.664,25.844, 

 cxs =1.529, 

 cxSR=0.510, 

 cxr =1.839, 

 cxRS=0.613, 

 SDIA=56.67,57.12,56.19,53.87,50.18, 

 SDEA=61.81,64.00,65.53,66.40,66.61, 

 RDIA=54.07,54.53,53.00,49.49,43.99, 

 RDEA=61.13,62.89,64.01,64.49,64.33, 

 endstg=0., 

 &END 

 &DATAIN STAGE=5., 

 RWG=5*1.0,GAMG=5*1.4, 

 KOGV=1, 

 DR=14.006,4*16.10, 

 DT=25.844,4*24.434 

 cxs =3.337, 

 cxSR=1.112, 

 cxr =0.5, 

 cxRS=0.5, 

 SDIA=53.13,52.92,51.43,48.65,44.58, 

 SDEA=5*0.0, 

 RDIA=5*0.0, 

 RDEA=5*0.0, 

 endstg=1.,endjob=0., 

 &END 

 

 70 PERCENT SPEED 

 &DATAIN STAGE=1.0,PTPS=1.45, 

 RPM=2086.0,ENDSTG=0.0 &END 

 &DATAIN STAGE=2. &END 

 &DATAIN STAGE=3. &END 

 &DATAIN STAGE=4. &END 

 &DATAIN STAGE=5.,VCTD=-1.,ENDSTG=1.,endjob=0. &END 

 

 120 PERCENT SPEED 

 &DATAIN STAGE=1.0,PTPS=1.3, 

 RPM=3576.0,ENDSTG=0.0 &END 

 &DATAIN STAGE=2. &END 

 &DATAIN STAGE=3. &END 

 &DATAIN STAGE=4. &END 

 &DATAIN STAGE=5.,VCTD=-1.,ENDSTG=1.,ENDJOB=1. &END 
NASA/TM—2014-218301 23 

Appendix B.—Sample Output 

The purpose of this output listing is to provide the reader with an impression in regarding the type and 

the amounts of data extracted from this code, and perhaps also provide a means for users to confirm the 

authenticity of the code they acquired. The printout listed here contains both the stage-by-stage printout 

(five stages altogether), and the inter-stage sector-by-sector printout (only the Stage 1 printout is given 

here to save the space). Note also that only the Case 1, Point 1 results are printed, the remaining Points 

and Cases are omitted. And, the printout of Standard Options (the echoing of the input options chosen, 

printed before the actual output is given) is also omitted. The output would look like: 

 NASA TURBINE CODE AXOD2 

1 NASA/P&W (Four and 1/2)-Stage Fan-Drive Turbine (ref. TMX-3498) 

 100 PERCENT SPEED (MUST GO FIRST) 

 CASE 1. 1 

 STAGE PERFORMANCE 

 STAGE 1 STAGE 2 STAGE 3 STAGE 4 STAGE 5 STAGE 6 STAGE 7 STAGE 8 

 TT 0 518.7 480.3 447.1 423.6 409.5 

 PT 0 14.700 10.778 8.010 6.458 5.622 

 WG 0 12.681 12.681 12.681 12.681 12.681 

 DEL H 9.214 7.970 5.640 3.390 0.000 

 WRT/P 19.647 25.785 33.476 40.416 45.640 

 DH/T 0.01776 0.01659 0.01261 0.00800 0.00000 

 N/RT 130.845 135.975 140.935 144.793 0.000 

 ETA TT 0.87243 0.85036 0.88083 0.85951 0.00000 

 ETA TS 0.62994 0.68533 0.77928 0.78243 0.00000 

 ETA AT 0.68504 0.73438 0.83823 0.85614 0.00000 

 PT0/PS1 1.350 1.296 1.188 1.101 1.028 

 PT0/PS2 1.561 1.455 1.279 1.165 1.037 

 PT0/PTBAR 1.364 1.346 1.240 1.149 1.010 

 PTR1A/PS2 1.344 1.245 1.135 1.076 1.027 

 TTR1A/TT0 0.96240 0.96317 0.97019 0.98031 1.00000 

 TTBAR/TT0 0.92597 0.93085 0.94743 0.96665 1.00000 

 PS 1 10.887 8.302 6.742 5.852 5.421 

 TTR 1A 499.2 462.6 433.8 415.2 409.5 

 PTR 1A 12.640 9.206 7.091 5.940 5.568 

 PS 2 9.416 7.406 6.263 5.541 5.422 

 TTR 2 499.4 462.9 434.1 415.6 409.5 

 PTR 2 12.657 9.219 7.106 5.961 5.568 

 TT 2A 480.1 447.0 423.5 409.2 409.5 

 PT 2A 10.753 7.996 6.449 5.616 5.568 

 TTBAR 2A 480.3 447.1 423.6 409.5 409.5 

 PTBAR 2A 10.778 8.010 6.458 5.622 5.566 

 UP/VI 0.26902 0.31375 0.41264 0.54953 0.00000 

 UR/VI 0.23607 0.25866 0.31669 0.39097 0.00000 

 W.F. P 4.35210 3.48096 2.28838 1.29549 0.00000 

 W.F. R 5.65164 5.12159 3.88500 2.55929 0.00000 

 RX P 0.31191 0.29314 0.29222 0.35184 -0.00631 

 RX R 0.22516 0.11563 -0.01386 -0.04760 -0.01139 

 REACT P 0.46496 0.38062 0.34313 0.39371 0.00000 

 REACT R 0.37441 0.14597 -0.05097 -0.09617 0.00000 

 ALPHA 0 0.000 58.386 55.002 40.012 8.493 

 I STATOR 3.000 5.621 1.574 -10.745 -41.062 

 ALF 1 64.186 68.216 68.115 65.490 0.000 

 BET 1A 56.431 57.544 49.984 26.512 0.000 

 I ROTOR 6.880 4.219 -0.192 -21.551 0.000 

 BET 2 67.164 68.501 67.129 63.960 0.000 

 ALF 2A 58.386 55.002 40.012 8.493 0.000 

 DBET P 123.595 126.046 117.113 90.472 0.000 

 DBET R 117.935 122.565 117.465 104.737 0.000 

 M 1 0.65289 0.58976 0.47807 0.35846 0.19583 

 M1 RT 0.71772 0.68735 0.58885 0.46414 0.20454 

 MR 1A 0.44547 0.37502 0.26007 0.15786 0.19583 

 MR1A RT 0.53832 0.51286 0.42182 0.31038 0.20454 

 MR 2 0.63415 0.54024 0.40994 0.31246 0.19494 

 MR2 TIP 0.61745 0.54040 0.43539 0.35560 0.18403 

 E/TH CR 9.214 8.607 6.543 4.152 0.000 

 N/RTH CR 2980.0 3096.8 3209.8 3297.6 0.0 

 WRTHCRE/D 12.678 16.639 21.601 26.079 29.451 

 RPM 2980.0 2980.0 2980.0 2980.0 0.0 

 MF 2A 0.22233 0.18561 0.15196 0.13441 0.19494 
NASA/TM—2014-218301 24 

 M 2A 0.42414 0.32362 0.19841 0.13591 0.19494 

 OVERALL PERFORMANCE 

 W.F. P 2.76278 W.F. R 4.39622 DEL H 26.21400 WNE/60D 629.646 TORQUE 828.694 

 WRT/P 19.64674 N/RT 130.845 DH/T 0.05054 N/RTH CR 2979.978 TOR/P 56.37376 

 PT0/PT2 2.64112 PT0/PS2 2.71114 PT/PAT2 2.64024 E/TH CR 26.21361 EQ TOR 2166.15792 

 ETA TT 0.86917 ETA TS 0.84939 ETA TTRP 0.86917 HP 470.204 U/VIS 0.15671 

 PT/T EQ 2.64112 PT/S EQ 2.71114 

+ WG 0 12.6809 EQ WG0 12.67753 PT0/PS1 1.35000 

1 NASA/P&W (Four and 1/2)-Stage Fan-Drive Turbine (ref. TMX-3498) 

 100 PERCENT SPEED (MUST GO FIRST) 

 CASE 1. 1 

 INTER-STAGE PERFORMANCE 

 STA 0 STATOR INLET STAGE 1 

 DIAM 0 15.956 16.260 16.869 17.478 18.087 18.696 19.000 

 SLOPE 0 -4.54 -1.99 0.57 3.13 5.67 

 WG 0 2.536 2.536 2.536 2.536 2.536 

 TT 0 518.7 518.7 518.7 518.7 518.7 518.7 518.7 

 PT 0 14.700 14.700 14.700 14.700 14.700 14.700 14.700 

 ALPHA 0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 

 I STA 0 2.350 2.481 2.745 2.978 3.152 3.235 3.276 

 V 0 295.948 295.948 295.948 295.948 295.948 295.948 295.948 

 VU 0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 

 VZ 0 295.948 295.948 295.948 295.948 295.948 295.948 295.948 

 TS 0 511.4 511.4 511.4 511.4 511.4 511.4 511.4 

 PS 0 13.990 13.990 13.990 13.990 13.990 13.990 13.990 

 DENS 0 0.07384 0.07384 0.07384 0.07384 0.07384 0.07384 0.07384 

 M 0 0.26696 0.26696 0.26696 0.26696 0.26696 0.26696 0.26696 

 STA 1 STATOR EXIT 

 DIAM 1 15.700 16.058 16.774 17.490 18.206 18.922 19.280 

 SLOPE 1 -3.89 0.53 4.94 9.29 13.54 

 WG 1 2.758 2.618 2.501 2.421 2.382 

 TT 1 518.7 518.7 518.7 518.7 518.7 518.7 518.7 

 ALF 1 58.623 59.892 62.428 64.186 65.182 65.424 65.545 

 DALF 58.623 59.892 62.428 64.186 65.182 65.424 65.545 

 V 1 763.107 750.130 724.174 699.706 677.670 657.255 647.048 

 VU 1 652.416 648.920 641.929 629.881 615.083 597.713 589.027 

 VZ 1 396.843 376.293 335.194 304.693 284.445 273.356 267.812 

 TS 1 470.3 471.9 475.1 478.0 480.5 482.7 483.9 

 PS 1 10.294 10.415 10.659 10.887 11.091 11.281 11.376 

 DENS 1 0.05909 0.05958 0.06056 0.06148 0.06231 0.06307 0.06346 

 M 1 0.71777 0.70444 0.67778 0.65289 0.63066 0.61022 0.60001 

 ZWI INC 0.9172 0.9016 0.8703 0.8478 0.8391 0.8474 0.8516 

 ETASC 0.9653 0.9633 0.9594 0.9560 0.9535 0.9523 0.9517 

 STATOR FORCES 

 AVG DIA 15.828 16.159 16.822 17.484 18.146 18.809 19.140 

 FTAN/IN -171.7 -166.9 -157.3 -147.9 -140.2 -134.4 -131.5 

 F TAN -247.5 

 FAX/IN 165.4 166.8 169.6 169.2 166.8 160.4 157.2 

 F AX 275.5 

 F DRUM -77.3 -106.7 

 STA 1A ROTOR INLET STAGE 1 

 DIAM 1A 15.538 15.944 16.755 17.567 18.379 19.190 19.596 

 SLOP 1A -3.89 0.53 4.94 9.29 13.54 

 WG 1A 2.758 2.618 2.501 2.421 2.382 

 TTR 1A 499.9 499.7 499.3 499.2 499.2 499.4 499.5 

 PTR 1A 12.686 12.672 12.646 12.640 12.651 12.680 12.695 

 BET 1A 52.500 53.476 55.428 56.431 56.445 55.360 54.817 

 I ROTOR 7.701 7.513 7.137 6.880 6.646 6.308 6.139 
NASA/TM—2014-218301 25 

 R 1A 575.036 555.315 515.872 478.506 444.391 413.054 397.385 

 RU 1A 456.996 446.256 424.777 398.702 370.336 339.835 324.585 

 MR 1A 0.53956 0.52028 0.48174 0.44547 0.41258 0.38253 0.36750 

 U 1A 202.036 207.312 217.865 228.418 238.971 249.524 254.801 

 STA 2 ROTOR EXIT 

 DIAM 2 15.538 16.000 16.923 17.846 18.769 19.692 20.154 

 SLOPE 2 -0.11 3.71 7.49 11.21 14.84 

 WG 2 2.579 2.505 2.478 2.509 2.610 

 BET 2 63.601 64.459 66.175 67.164 67.430 66.967 66.736 

 DBET 116.101 117.935 121.603 123.595 123.875 122.327 121.553 

 R 2 706.926 698.391 681.320 668.349 659.200 654.136 651.604 

 RU 2 633.583 630.142 623.258 615.961 608.713 601.988 598.626 

 MR 2 0.67356 0.66485 0.64742 0.63415 0.62475 0.61943 0.61677 

 U 2 202.036 208.038 220.042 232.046 244.050 256.054 262.056 

 RX 0.23580 0.25100 0.28139 0.31191 0.34045 0.36833 0.38227 

 DEL H 8.811 8.919 9.136 9.280 9.370 9.412 9.432 

 PSI 2.69044 2.58882 2.38559 2.19153 2.01108 1.84373 1.76005 

 ETA TT 0.86694 0.86810 0.87040 0.87239 0.87525 0.87984 0.88213 

 ETA TS 0.56830 0.58295 0.61226 0.63602 0.65541 0.67078 0.67846 

 ETA AT 0.63521 0.64597 0.66748 0.68718 0.70602 0.72449 0.73373 

 ZWI INC 1.3999 1.3660 1.2983 1.2554 1.2408 1.2543 1.2611 

 ETARC 0.9328 0.9296 0.9232 0.9187 0.9167 0.9176 0.9180 

 ROTOR FORCES 

 AVG DIA 15.538 15.972 16.839 17.706 18.574 19.441 19.875 

 FTAN/IN 204.4 198.8 187.5 179.2 174.7 174.3 174.2 

 F TAN 397.2 

 FAX/IN 67.8 74.8 83.3 92.8 102.8 112.8 116.1 

 F AX 201.9 

 F DRUM 0.0 -183.6 

 STA 2A STAGE EXIT STAGE 1 

 DIAM 2A 15.458 15.961 16.968 17.975 18.982 19.989 20.492 

 SLOP 2A -0.11 3.71 7.49 11.21 14.84 

 WG 2A 2.579 2.505 2.478 2.509 2.610 

 PT 2A 10.910 10.873 10.799 10.753 10.730 10.733 10.734 

 TT 2A 482.0 481.6 480.7 480.1 479.7 479.5 479.4 

 V 2A 520.163 504.631 473.567 447.584 426.539 410.890 403.066 

 VU 2A 433.602 423.114 402.138 381.160 360.578 340.808 330.923 

 ALF 2A 56.407 56.978 58.121 58.386 57.710 56.041 55.206 

 MF 2A 0.27351 0.26146 0.23736 0.22233 0.21566 0.21703 0.21771 

 M 2A 0.49495 0.47978 0.44944 0.42414 0.40370 0.38852 0.38093 

 VZ 2A 287.454 275.003 250.101 234.624 227.857 229.523 230.355 

 TS 2A 459.5 460.4 462.0 463.4 464.5 465.4 465.9 

 PS 2A 9.232 9.288 9.402 9.502 9.591 9.672 9.713 

 DENS 2A 0.05423 0.05446 0.05493 0.05534 0.05572 0.05609 0.05627 

 

NASA/TM—2014-218301 27 

Appendix C.—Sample Plot File 

The following is the plot file generated. Two parts are created in each plot file. For this example, they 

are: pressure ratio (PR) versus efficiency (EFF), and PR versus mass flow rate (FLOW). But other choices 

are available, for example, power output (HP), or torque (TORQ) can replace “FLOW”. The options of 

plot, ‘iplot(3)’, given in Section 3.0 of this paper control the combination. 

 269 TURBINE EFFICIENCY VS. PR, RPM, AND AREA 

AREA 1 1.00 

SPED 3 70.0 100.0 120.0 

PR 88 2.3421 2.3521 2.3623 2.3726 2.3831 2.3937 2.4044 

PR 88 2.4154 2.4265 2.4377 2.4492 2.4608 2.4727 2.4848 

PR 88 2.4970 2.5095 2.5223 2.5353 2.5485 2.5621 2.5759 

PR 88 2.5900 2.6044 2.6193 2.6344 2.6500 2.6659 2.6823 

PR 88 2.6991 2.7165 2.7343 2.7528 2.7718 2.7916 2.8120 

PR 88 2.8332 2.8553 2.8783 2.9023 2.9275 2.9539 2.9818 

PR 88 3.0112 3.0424 3.0757 3.1114 3.1498 3.1916 3.2373 

PR 88 3.2879 3.3446 3.4092 3.4846 3.5750 3.6890 3.8441 

PR 88 4.0892 4.7206 5.0386 5.8059 5.8403 5.8735 5.9057 

PR 88 5.9371 5.9677 5.9975 6.0267 6.0551 6.0828 6.1099 

PR 88 6.1362 6.1617 6.1863 6.2098 6.2327 6.2551 6.2769 

PR 88 6.2982 6.3189 6.3390 6.3585 6.3773 6.3954 6.4129 

PR 88 6.4298 6.4461 6.4620 6.4776 

EFF 88 0.81295 0.81254 0.81214 0.81173 0.81132 0.81090 0.81049 

EFF 88 0.81007 0.80964 0.80922 0.80879 0.80835 0.80792 0.80747 

EFF 88 0.80703 0.80658 0.80612 0.80567 0.80520 0.80473 0.80426 

EFF 88 0.80378 0.80329 0.80280 0.80230 0.80179 0.80128 0.80076 

EFF 88 0.80023 0.79969 0.79914 0.79858 0.79801 0.79742 0.79683 

EFF 88 0.79622 0.79559 0.79495 0.79430 0.79362 0.79292 0.79219 

EFF 88 0.79144 0.79067 0.78985 0.78900 0.78810 0.78715 0.78614 

EFF 88 0.78505 0.78387 0.78258 0.78113 0.77947 0.77749 0.77499 

EFF 88 0.77143 0.76389 0.76074 0.75377 0.75336 0.75294 0.75251 

EFF 88 0.75205 0.75157 0.75107 0.75056 0.75003 0.74949 0.74894 

EFF 88 0.74839 0.74784 0.74731 0.74680 0.74630 0.74578 0.74526 

EFF 88 0.74473 0.74421 0.74369 0.74317 0.74266 0.74216 0.74166 

EFF 88 0.74118 0.74070 0.74022 0.73973 

PR 89 2.6411 2.6598 2.6787 2.6981 2.7178 2.7379 2.7584 

PR 89 2.7793 2.8007 2.8225 2.8449 2.8677 2.8911 2.9151 

PR 89 2.9397 2.9649 2.9908 3.0175 3.0449 3.0731 3.1022 

PR 89 3.1323 3.1633 3.1955 3.2288 3.2633 3.2993 3.3367 

PR 89 3.3757 3.4165 3.4593 3.5042 3.5516 3.6017 3.6548 

PR 89 3.7113 3.7719 3.8369 3.9073 3.9840 4.0683 4.1620 

PR 89 4.2672 4.3877 4.5287 4.6986 4.9139 5.2104 5.7124 

PR 89 6.5416 6.5943 6.6448 6.6934 6.7403 6.7856 6.8296 

PR 89 6.8722 6.9138 6.9543 6.9939 7.0326 7.0706 7.1079 

PR 89 7.1446 7.1805 7.2157 7.2503 7.2840 7.3169 7.3489 

PR 89 7.3798 7.4101 7.4398 7.4689 7.4972 7.5248 7.5515 

PR 89 7.5773 7.6023 7.6262 7.6493 7.6717 7.6935 7.7147 

PR 89 7.7353 7.7553 7.7747 7.7934 7.8114 

EFF 89 0.86917 0.86917 0.86917 0.86916 0.86915 0.86914 0.86913 

EFF 89 0.86911 0.86909 0.86907 0.86905 0.86903 0.86900 0.86897 

EFF 89 0.86893 0.86890 0.86886 0.86881 0.86877 0.86872 0.86867 

EFF 89 0.86861 0.86855 0.86849 0.86842 0.86834 0.86826 0.86818 

EFF 89 0.86808 0.86798 0.86787 0.86776 0.86763 0.86749 0.86734 

EFF 89 0.86718 0.86700 0.86680 0.86658 0.86634 0.86607 0.86576 

EFF 89 0.86540 0.86499 0.86450 0.86390 0.86312 0.86202 0.86005 

EFF 89 0.85595 0.85557 0.85520 0.85482 0.85443 0.85402 0.85360 

EFF 89 0.85315 0.85269 0.85219 0.85167 0.85113 0.85056 0.84996 

EFF 89 0.84934 0.84871 0.84805 0.84739 0.84672 0.84605 0.84538 

EFF 89 0.84473 0.84406 0.84339 0.84271 0.84203 0.84135 0.84068 

EFF 89 0.84002 0.83937 0.83875 0.83813 0.83752 0.83690 0.83629 
NASA/TM—2014-218301 28 

EFF 89 0.83569 0.83508 0.83449 0.83390 0.83332 

PR 81 2.7314 2.7564 2.7820 2.8081 2.8349 2.8623 2.8903 

PR 81 2.9191 2.9486 2.9788 3.0099 3.0418 3.0747 3.1085 

PR 81 3.1434 3.1794 3.2165 3.2548 3.2945 3.3356 3.3783 

PR 81 3.4226 3.4688 3.5169 3.5671 3.6197 3.6748 3.7328 

PR 81 3.7939 3.8586 3.9273 4.0004 4.0787 4.1629 4.2541 

PR 81 4.3535 4.4629 4.5843 4.7213 4.8783 5.0622 5.2856 

PR 81 5.5720 5.9802 7.0350 7.1029 7.1681 7.2309 7.2914 

PR 81 7.3496 7.4058 7.4603 7.5131 7.5645 7.6146 7.6635 

PR 81 7.7114 7.7584 7.8045 7.8499 7.8946 7.9388 7.9823 

PR 81 8.0253 8.0676 8.1093 8.1501 8.1901 8.2292 8.2671 

PR 81 8.3044 8.3410 8.3768 8.4117 8.4458 8.4789 8.5108 

PR 81 8.5417 8.5714 8.6003 8.6236 

EFF 81 0.87251 0.87296 0.87340 0.87383 0.87425 0.87466 0.87507 

EFF 81 0.87548 0.87587 0.87627 0.87665 0.87704 0.87742 0.87780 

EFF 81 0.87818 0.87855 0.87892 0.87930 0.87967 0.88004 0.88042 

EFF 81 0.88080 0.88118 0.88156 0.88194 0.88234 0.88273 0.88313 

EFF 81 0.88354 0.88395 0.88438 0.88480 0.88524 0.88569 0.88616 

EFF 81 0.88662 0.88710 0.88758 0.88808 0.88858 0.88907 0.88955 

EFF 81 0.88999 0.89028 0.88884 0.88855 0.88825 0.88793 0.88760 

EFF 81 0.88726 0.88691 0.88654 0.88615 0.88574 0.88530 0.88482 

EFF 81 0.88432 0.88379 0.88322 0.88262 0.88198 0.88131 0.88061 

EFF 81 0.87988 0.87913 0.87836 0.87758 0.87679 0.87601 0.87524 

EFF 81 0.87446 0.87367 0.87288 0.87209 0.87130 0.87053 0.86977 

EFF 81 0.86903 0.86831 0.86759 0.98296 

EOT 

 270 TURBINE FLOW FUNCTION VS. PR, RPM, AND AREA 

AREA 1 1.00 

SPED 3 70.0 100.0 120.0 

PR 88 2.3421 2.3521 2.3623 2.3726 2.3831 2.3937 2.4044 

PR 88 2.4154 2.4265 2.4377 2.4492 2.4608 2.4727 2.4848 

PR 88 2.4970 2.5095 2.5223 2.5353 2.5485 2.5621 2.5759 

PR 88 2.5900 2.6044 2.6193 2.6344 2.6500 2.6659 2.6823 

PR 88 2.6991 2.7165 2.7343 2.7528 2.7718 2.7916 2.8120 

PR 88 2.8332 2.8553 2.8783 2.9023 2.9275 2.9539 2.9818 

PR 88 3.0112 3.0424 3.0757 3.1114 3.1498 3.1916 3.2373 

PR 88 3.2879 3.3446 3.4092 3.4846 3.5750 3.6890 3.8441 

PR 88 4.0892 4.7206 5.0386 5.8059 5.8403 5.8735 5.9057 

PR 88 5.9371 5.9677 5.9975 6.0267 6.0551 6.0828 6.1099 

PR 88 6.1362 6.1617 6.1863 6.2098 6.2327 6.2551 6.2769 

PR 88 6.2982 6.3189 6.3390 6.3585 6.3773 6.3954 6.4129 

PR 88 6.4298 6.4461 6.4620 6.4776 

FLOW 88 13.3185 13.3283 13.3380 13.3477 13.3572 13.3667 13.3760 

FLOW 88 13.3853 13.3944 13.4035 13.4125 13.4214 13.4302 13.4389 

FLOW 88 13.4475 13.4561 13.4645 13.4729 13.4812 13.4894 13.4975 

FLOW 88 13.5056 13.5135 13.5214 13.5292 13.5369 13.5445 13.5521 

FLOW 88 13.5596 13.5670 13.5743 13.5816 13.5887 13.5958 13.6029 

FLOW 88 13.6098 13.6167 13.6235 13.6303 13.6369 13.6435 13.6500 

FLOW 88 13.6565 13.6629 13.6692 13.6755 13.6816 13.6878 13.6938 

FLOW 88 13.6998 13.7057 13.7116 13.7174 13.7231 13.7288 13.7344 

FLOW 88 13.7399 13.7454 13.7463 13.7468 13.7468 13.7468 13.7468 

FLOW 88 13.7468 13.7468 13.7468 13.7468 13.7468 13.7468 13.7468 

FLOW 88 13.7468 13.7468 13.7468 13.7468 13.7468 13.7468 13.7468 

FLOW 88 13.7468 13.7468 13.7468 13.7468 13.7468 13.7468 13.7468 

FLOW 88 13.7468 13.7468 13.7468 13.7468 

PR 89 2.6411 2.6598 2.6787 2.6981 2.7178 2.7379 2.7584 

PR 89 2.7793 2.8007 2.8225 2.8449 2.8677 2.8911 2.9151 

PR 89 2.9397 2.9649 2.9908 3.0175 3.0449 3.0731 3.1022 

PR 89 3.1323 3.1633 3.1955 3.2288 3.2633 3.2993 3.3367 

PR 89 3.3757 3.4165 3.4593 3.5042 3.5516 3.6017 3.6548 

PR 89 3.7113 3.7719 3.8369 3.9073 3.9840 4.0683 4.1620 

PR 89 4.2672 4.3877 4.5287 4.6986 4.9139 5.2104 5.7124 

PR 89 6.5416 6.5943 6.6448 6.6934 6.7403 6.7856 6.8296 
NASA/TM—2014-218301 29 

PR 89 6.8722 6.9138 6.9543 6.9939 7.0326 7.0706 7.1079 

PR 89 7.1446 7.1805 7.2157 7.2503 7.2840 7.3169 7.3489 

PR 89 7.3798 7.4101 7.4398 7.4689 7.4972 7.5248 7.5515 

PR 89 7.5773 7.6023 7.6262 7.6493 7.6717 7.6935 7.7147 

PR 89 7.7353 7.7553 7.7747 7.7934 7.8114 

FLOW 89 12.6809 12.6969 12.7128 12.7285 12.7441 12.7595 12.7748 

FLOW 89 12.7899 12.8049 12.8198 12.8345 12.8490 12.8634 12.8777 

FLOW 89 12.8918 12.9058 12.9196 12.9334 12.9470 12.9604 12.9737 

FLOW 89 12.9869 13.0000 13.0129 13.0258 13.0384 13.0510 13.0635 

FLOW 89 13.0758 13.0880 13.1001 13.1120 13.1239 13.1356 13.1472 

FLOW 89 13.1588 13.1702 13.1814 13.1926 13.2037 13.2146 13.2255 

FLOW 89 13.2362 13.2469 13.2574 13.2678 13.2782 13.2884 13.2985 

FLOW 89 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 

FLOW 89 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 

FLOW 89 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 

FLOW 89 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 

FLOW 89 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 13.3034 

FLOW 89 13.3034 13.3034 13.3034 13.3034 13.3034 

PR 81 2.7314 2.7564 2.7820 2.8081 2.8349 2.8623 2.8903 

PR 81 2.9191 2.9486 2.9788 3.0099 3.0418 3.0747 3.1085 

PR 81 3.1434 3.1794 3.2165 3.2548 3.2945 3.3356 3.3783 

PR 81 3.4226 3.4688 3.5169 3.5671 3.6197 3.6748 3.7328 

PR 81 3.7939 3.8586 3.9273 4.0004 4.0787 4.1629 4.2541 

PR 81 4.3535 4.4629 4.5843 4.7213 4.8783 5.0622 5.2856 

PR 81 5.5720 5.9802 7.0350 7.1029 7.1681 7.2309 7.2914 

PR 81 7.3496 7.4058 7.4603 7.5131 7.5645 7.6146 7.6635 

PR 81 7.7114 7.7584 7.8045 7.8499 7.8946 7.9388 7.9823 

PR 81 8.0253 8.0676 8.1093 8.1501 8.1901 8.2292 8.2671 

PR 81 8.3044 8.3410 8.3768 8.4117 8.4458 8.4789 8.5108 

PR 81 8.5417 8.5714 8.6003 8.6236 

FLOW 81 12.2239 12.2445 12.2648 12.2849 12.3049 12.3246 12.3441 

FLOW 81 12.3635 12.3826 12.4016 12.4204 12.4390 12.4574 12.4756 

FLOW 81 12.4937 12.5115 12.5292 12.5467 12.5641 12.5813 12.5983 

FLOW 81 12.6151 12.6318 12.6483 12.6647 12.6809 12.6969 12.7128 

FLOW 81 12.7285 12.7441 12.7595 12.7748 12.7899 12.8049 12.8198 

FLOW 81 12.8345 12.8490 12.8634 12.8777 12.8918 12.9058 12.9196 

FLOW 81 12.9334 12.9470 12.9598 12.9598 12.9598 12.9598 12.9598 

FLOW 81 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 

FLOW 81 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 

FLOW 81 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 

FLOW 81 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 12.9598 

FLOW 81 12.9598 12.9598 12.9598 12.9598 

EOT 

NASA/TM—2014-218301 31 

Appendix D.—Sample Results 

The results of this sample case were published in Reference 4. They are echoed here for illustration 

purpose and for completeness. 

Figure 2.—Comparison of the equivalent mass flow rate (Ref. 8). 

Figure 3.—Comparison of the equivalent torque (Ref. 8). 

12.2

12.4

12.6

12.8

13

13.2

13.4

13.6

13.8

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Equivalent Mass Flow (lbm/s) 

Total-to-Total Pressure Ratio 

NASA/PW FAN-DRIVE TURBINE (4 +OGV stages) 

FLOW MAP 

Lines -- Prediction by AXOD 

Symbols -- Experimental Data 

 70% Speed 

100% Speed 

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Equivalent Torque (ft-lb) 

Total-to-Total Pressure Ratio 

NASA/PW FAN-DRIVE TURBINE (4 +OGV stages) 

TORQUE MAP 

Lines -- Prediction by AXOD 

Symbols -- Experimental Data 

70% Speed 

100% Speed 
NASA/TM—2014-218301 32 

Figure 4.—Comparison of the total efficiency (Ref. 8). 

0.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9

Total Efficiency 

Total-to-Total Pressure Ratio 

NASA/PW FAN-DRIVE TURBINE (4 +OGV stages) 

EFFICIENCY MAP 

Lines -- Prediction by AXOD 

Symbols -- Experimental Data 

70% Speed 

100% Speed 

D 

D -- The Design Point Conceived 
NASA/TM—2014-218301 33 

References 

1. Flagg, E. E., “Analytical Procedure and Computer Program for Determining the Off-design 

Performance of Axial Flow Turbines,” NASA CR-710, February 1967. 

2. Glassman, A. J., “Modeling Improvements and User’s Manual for Axial-Flow Turbine Off-Design 

Computer Code AXOD,” NASA CR 195370, August 1994. 

3. Chen, S. S., “Preliminary Axial Flow Turbine Design and Off-Design Performance Analysis Methods 

for Rotary Wing Aircraft Engines; I- Validation,” NASA/TM—2009-215651/PART1, May 2009. 

4. Chen, S. S., “Capability Extension to the Turbine Off-Design Computer Program AXOD With 

Applications to the Highly Loaded Fan-Drive Turbines,” NASA/TM—2011-217129, July 2011. 

5. Glassman, A. J., ed., Turbine Design and Application; Volume I, NASA SP-290, 1972. 

6. Glassman, A. J., “Computer Code for Preliminary Sizing Analysis of Axial-Flow Turbines,” NASA 

CR 4430, February 1992. 

7. Webster, P. F., “Design of a 4 1⁄2–Stage Turbine with a Stage Loading Factor of 4.66 and High 

Specific Work Output,” NASA CR-2659, March 1976. 

8. Whitney, W. J., Behning, F. P., Moffitt, T. P., and Hutz, G. M., “Cold-Air Investigation of 4 1⁄2–Stage 

Turbine with Stage Loading Factor of 4.66 and High Specific Work Output; I- Overall Performance,” 

NASA TM X-3498, February 1977.